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WASHINGTON — A bipartisan group of six senators is closing in on what could represent the best chance for tackling a deficit crisis that has forced the government to borrow more than 40 cents of every dollar it spends.

Their plan, still a work in progress, would reduce borrowing by up to $4 trillion over the next decade by putting the two parties’ sacred cows on the chopping block. Republicans would have to agree to higher taxes while Democrats would have to accept cuts in popular benefit programs such as Medicare, Medicaid, and maybe even Social Security.

There is urgency to their work.

Most Republicans and some Democrats in Congress have said they will not vote to increase the government’s ability to borrow without some action addressing the nation’s long-term debt.

The government is expected to reach its borrowing limit of $14.3 trillion by mid-May. Treasury Secretary Timothy Geithner says steps are being taken to delay until July what would be an unprecedented default on the debt.

Geithner and a growing number of business leaders say a US default would plunge the United States, and perhaps the world, into a second economic crisis.

Despite opposition from top leaders in both parties, the bipartisan group of senators is banking on support from many of the 64 senators who wrote President Obama in March urging him to support a comprehensive deficit reduction effort.

“There’s a huge appetite for a starting point that’s bipartisan,’’ said Senator Mark Warner, a Virginia Democrat, one of the six senators, three from each party. While some might cling to the belief that the issue can be put off until after the next presidential election, he said, “I think there’s a growing consensus we may not have that long.’’

The group includes four members of Obama’s deficit commission, and its recommendations are expected to closely track the commission’s plan for $3 in spending cuts for every $1 in tax revenue increases over the coming decade.

Its other two Democrats are Senator Dick Durbin of Illinois, the second-ranking Democratic leader, and Senator Kent Conrad of North Dakota, the Budget Committee chairman. The Republicans are Senators Tom Coburn of Oklahoma, Mike Crapo of Idaho, and Saxby Chambliss of Georgia.

The six have met in private for several months, even as House Republicans and Obama developed more partisan plans that have little chance of being enacted into law because of Washington’s divided government.

“We’ve made enormous progress in that group. And I hope that we are able to announce an agreement soon,’’ Conrad said on “Fox News Sunday.’’ He added: “If we don’t, we’re simply not going to be relevant because this debate marches on.’’

House Republicans passed a nonbinding plan in April that calls for reducing annual deficits by a total of $6.2 trillion over the next decade. It includes no tax increases but calls for transforming Medicare from a program in which the government directly pays medical bills into a voucherlike system that subsidizes the purchase of private insurance plans.

Obama has outlined a plan to reduce borrowing by $4 trillion over the next 12 years. His plan includes $1 trillion in tax increases and is less specific about how he would cut benefit programs.

Vice President Joe Biden is to begin leading a series of bipartisan talks this week on reducing the debt. The group of six senators wants its plan to be part of the discussion.

The senators’ work is rooted in a simple political reality: Getting anything actually passed into law given the present balance of power in Washington requires both Democrats and Republicans to embrace proposals that make each uncomfortable. An approach that leaves politically challenging topics off the table simply won’t make a dent in deficits averaging $1 trillion a year or so over the upcoming decade.

“A Republican plan will not pass. A Democratic plan will not pass,’’ Chambliss said. “It is going to require locking arms and jumping off the building together.’’

The plan has one important advantage: It’s the only truly bipartisan effort in a Senate teeming with politicians promising to tackle the debt but lacking the trust needed to take politically poisonous positions such as cutting federal benefit programs and boosting tax revenues.

The goal is to release the plan within the next few weeks.

“I think a lot of people will be surprised at the support we get — from all corners,’’ Warner said.

For Republicans, there’s enormous political risk in embracing higher revenues, even if they come in the form of a tax overhaul that lowers overall rates while raising money through reducing credits, deductions, and exemptions. Democrats face huge risks of their own, especially if they endorse politically dangerous cuts to Social Security.

Obama has shied away from tackling Social Security’s problems, and top Democrats in Congress, including Senate Majority Leader Harry Reid of Nevada and House Minority Leader Nancy Pelosi of California, have ruled out cuts in benefits, including changes that would only affect future retirees, such as raising the retirement age.

The plan by Obama’s deficit commission, which the president did not endorse, would have nearly frozen the Pentagon’s budget and cut spending at most domestic agencies. It called for gradually raising the retirement age for full Social Security benefits to 69 and curbing future cost-of-living increases. The full retirement age is 66, and is gradually increasing to 67 for people born in 1960 or later.